-
Previous Article
Fully discrete finite element approximation of the 2D/3D unsteady incompressible magnetohydrodynamic-Voigt regularization flows
- DCDS-B Home
- This Issue
-
Next Article
Stabilities and dynamic transitions of the Fitzhugh-Nagumo system
Well-posedness and decay of solutions to 3D generalized Navier-Stokes equations
1. | College of of Sciences, Northeastern University, Shenyang 110819, P. R. China |
2. | School of Mathematics (Zhuhai), Sun Yat-Sen University, Zhuhai 519082, P. R. China |
The global well-posedness and large time behavior of solutions for the Cauchy problem of the three-dimensional generalized Navier-Stokes equations are studied. We first construct a local continuous solution, then by combining some a priori estimates and the continuity argument, the local continuous solution is extended to all $ t>0 $ step by step provided that the initial data is sufficiently small. In addition, by using Strauss's inequality, generalized interpolation type lemma and a bootstrap argument, we establish the $ L^p $ decay estimate for the solution $ u(\cdot,t) $ and all its derivatives for generalized Navier-Stokes equations with $ \max\{1,\frac{3+q}6\}<\alpha\leq\frac12+\min\{\frac3q-\frac3p,\frac3{2p}\} $.
References:
[1] |
L. Caffarelli, R. Kohn and L. Nirenberg,
Partial regularity of suitable weak solutions of the Navier-Stokes equations, Comm. Pure Appl. Math., 35 (1982), 771-831.
doi: 10.1002/cpa.3160350604. |
[2] |
D. Chae and J. Lee, On the global well-posedness and stability of the Navier-Stokes and the related equations,, [Contributions to Current Challenges in Mathematical Fluid Mechanics], in Adv. Math. Fluid Mech., Birkhäuser, Basel, (2004), 31–51. |
[3] |
P. Constantin and J. Wu,
Behavior of solutions of 2D quasi-geostrophic equations, SIAM J. Math. Anal., 30 (1999), 937-948.
doi: 10.1137/S0036141098337333. |
[4] |
X. Ding and J. Wang,
Global solution for a semilinear parabolic system, Acta Math. Sci. (English Ed.), 3 (1983), 397-414.
doi: 10.1016/S0252-9602(18)30621-0. |
[5] |
J. Fan, Y. Fukumoto and Y. Zhou,
Logarithmically improved regularity criteria for the generalized Navier-Stokes and related equations, Kinet. Relat. Models, 6 (2013), 545-556.
doi: 10.3934/krm.2013.6.545. |
[6] |
A. Friedman, Partial Differential Equations, Holt, Reinhart and Winston, New York, 1969. |
[7] |
D. Hoff and J. A. Smooler,
Global existence for systems of parabolic conservation laws in several space variables, J. Differential Equations, 68 (1987), 210-220.
doi: 10.1016/0022-0396(87)90192-6. |
[8] |
D. Hoff and J. A. Smooler,
Solutions in the large for certain nonlinear parabolic systems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 2 (1985), 213-235.
doi: 10.1016/S0294-1449(16)30403-6. |
[9] |
Z. Jiang,
Asymptotic behavior of strong solutions to the 3D Navier-Stokes equations with a nonlinear damping term, Nonlinear Anal., 75 (2012), 5002-5009.
doi: 10.1016/j.na.2012.04.014. |
[10] |
Z. Jiang and J. Fan,
Time decay rate for two 3D magnetohydrodynamics-$\alpha$ models, Math. Methods Appl. Sci., 37 (2014), 838-845.
doi: 10.1002/mma.2840. |
[11] |
Q. Jiu and H. Yu,
Decay of solutions to the three-dimensional generalized Navier-Stokes equations, Asymptot. Anal., 94 (2015), 105-124.
doi: 10.3233/ASY-151307. |
[12] |
T. Kato,
Strong $L^p$-solutions of the Navier-Stokes equation in $\mathbb{R}^m$, with applications to weak solutions, Math. Z., 187 (1984), 471-480.
doi: 10.1007/BF01174182. |
[13] |
I. Kukavica,
Space-time decay for solutions of the Navier-Stokes equations, Indiana Univ. Math. J., 50 (2001), 205-222.
doi: 10.1512/iumj.2001.50.2084. |
[14] |
I. Kukavica and J. J. Torres,
Weighted bounds for the velocity and the vorticity for the Navier-Stokes equations, Nonlinearity, 19 (2006), 293-303.
doi: 10.1088/0951-7715/19/2/003. |
[15] |
N. H. Katz and N. Pavlović,
A cheap Caffarelli-Kohn-Nirenberg inequality for the Navier-Stokes equation with hyperdissipation, Geom. Funct. Anal., 12 (2002), 355-379.
doi: 10.1007/s00039-002-8250-z. |
[16] |
H.-O. Kreiss, T. Hagstrom, J. Lorenz and P. Zingano,
Decay in time of incompressible flows, J. Math. Fluid Mech., 5 (2003), 231-244.
doi: 10.1007/s00021-003-0079-1. |
[17] |
J. Leray, Étude de diverses équations integrales non lineaires et de quelques problémes que pose l'hydrodynamique, Thèses de l'entre-deux-guerres, 142 (1933), 88pp. |
[18] |
J. Leray,
Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math., 63 (1934), 193-248.
doi: 10.1007/BF02547354. |
[19] |
P. Li and Z. Zhai,
Well-posedness and regularity of generalized Navier-Stokes equations in some critical $Q$-spaces, J. Funct. Anal., 259 (2010), 2457-2519.
doi: 10.1016/j.jfa.2010.07.013. |
[20] |
J.-L. Lions, Quelques Méthodes De Résolution Des Problémes Aux Limites Non Linéaires, Dunod, Gauthier-Villars, Paris, 1969. |
[21] |
Q. Liu and J. Zhao,
Global well-posedness for the generalized magneto-hydrodynamic equations in the critical Fourier-Herz spaces, J. Math. Anal. Appl., 420 (2014), 1301-1315.
doi: 10.1016/j.jmaa.2014.06.031. |
[22] |
Q. Liu, J. Zhao and S. Cui,
Existence and regularizing rate estimates of solutions to a generalized magneto-hydrodynamic system in pseudomeasure spaces, Ann. Mat. Pura Appl. (4), 191 (2012), 293-309.
doi: 10.1007/s10231-010-0184-8. |
[23] |
S. Liu, F. Wang and H. Zhao,
Global existence and asymptotics of solutions of the Cahn-Hilliard equation, J. Differential Equations, 238 (2007), 426-469.
doi: 10.1016/j.jde.2007.02.014. |
[24] |
C. Miao, B. Yuan and B. Zhang,
Well-posedness of the Cauchy problem for the fractional power dissipative equations, Nonlinear Anal., 68 (2008), 461-484.
doi: 10.1016/j.na.2006.11.011. |
[25] |
M. E. Schonbek,
$L^2$ decay for weak solutions of the Navier-Stokes equations, Arch. Ration. Mech. Anal., 88 (1985), 209-222.
doi: 10.1007/BF00752111. |
[26] |
M. E. Schonbek,
Large time behaviour of solutions to the Navier-Stokes equations, Comm. Partial Differential Equations, 11 (1986), 733-763.
doi: 10.1080/03605308608820443. |
[27] |
E. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, USA, 1970.
![]() ![]() |
[28] |
W. A. Strauss,
Decay and asymptotic for $u_tt-\Delta u=F(u)$, J. Funct. Anal., 2 (1968), 409-457.
doi: 10.1016/0022-1236(68)90004-9. |
[29] |
S. Weng,
Remarks on asymptotic behaviors of strong solutions to a viscous Boussinesq system, Math. Method Appl. Sci., 39 (2016), 4398-4418.
doi: 10.1002/mma.3868. |
[30] |
M. Wiegner,
Decay results for weak solutions of the Navier-Stokes equations on $\mathbb{R}^n$, J. London Math. Soc., 35 (1987), 303-313.
doi: 10.1112/jlms/s2-35.2.303. |
[31] |
J. Wu,
Generalized MHD equations, J. Differential Equations, 195 (2003), 284-312.
doi: 10.1016/j.jde.2003.07.007. |
[32] |
J. Wu,
Regularity criteria for the generalized MHD equations, Comm. Partial Differential Equations, 33 (2008), 285-306.
doi: 10.1080/03605300701382530. |
[33] |
Z. Ye,
Global well-posedness and decay results to 3D generalized viscous magnetohydrodynamic equations, Ann. Mat. Pura Appl. (4), 195 (2016), 1111-1121.
doi: 10.1007/s10231-015-0507-x. |
[34] |
Y. Zhou,
Regularity criteria for the generalized viscous MHD equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 491-505.
doi: 10.1016/j.anihpc.2006.03.014. |
[35] |
Y. Zhou,
A remark on the decay of solutions to the 3-D Navier-Stokes equations, Math. Methods Appl. Sci., 30 (2007), 1223-1229.
doi: 10.1002/mma.841. |
[36] |
Y. Zhou,
Asymptotic behaviour of the solutions to the 2D dissipative quasi-geostrophic flows, Nonlinearity, 21 (2008), 2061-2071.
doi: 10.1088/0951-7715/21/9/008. |
show all references
References:
[1] |
L. Caffarelli, R. Kohn and L. Nirenberg,
Partial regularity of suitable weak solutions of the Navier-Stokes equations, Comm. Pure Appl. Math., 35 (1982), 771-831.
doi: 10.1002/cpa.3160350604. |
[2] |
D. Chae and J. Lee, On the global well-posedness and stability of the Navier-Stokes and the related equations,, [Contributions to Current Challenges in Mathematical Fluid Mechanics], in Adv. Math. Fluid Mech., Birkhäuser, Basel, (2004), 31–51. |
[3] |
P. Constantin and J. Wu,
Behavior of solutions of 2D quasi-geostrophic equations, SIAM J. Math. Anal., 30 (1999), 937-948.
doi: 10.1137/S0036141098337333. |
[4] |
X. Ding and J. Wang,
Global solution for a semilinear parabolic system, Acta Math. Sci. (English Ed.), 3 (1983), 397-414.
doi: 10.1016/S0252-9602(18)30621-0. |
[5] |
J. Fan, Y. Fukumoto and Y. Zhou,
Logarithmically improved regularity criteria for the generalized Navier-Stokes and related equations, Kinet. Relat. Models, 6 (2013), 545-556.
doi: 10.3934/krm.2013.6.545. |
[6] |
A. Friedman, Partial Differential Equations, Holt, Reinhart and Winston, New York, 1969. |
[7] |
D. Hoff and J. A. Smooler,
Global existence for systems of parabolic conservation laws in several space variables, J. Differential Equations, 68 (1987), 210-220.
doi: 10.1016/0022-0396(87)90192-6. |
[8] |
D. Hoff and J. A. Smooler,
Solutions in the large for certain nonlinear parabolic systems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 2 (1985), 213-235.
doi: 10.1016/S0294-1449(16)30403-6. |
[9] |
Z. Jiang,
Asymptotic behavior of strong solutions to the 3D Navier-Stokes equations with a nonlinear damping term, Nonlinear Anal., 75 (2012), 5002-5009.
doi: 10.1016/j.na.2012.04.014. |
[10] |
Z. Jiang and J. Fan,
Time decay rate for two 3D magnetohydrodynamics-$\alpha$ models, Math. Methods Appl. Sci., 37 (2014), 838-845.
doi: 10.1002/mma.2840. |
[11] |
Q. Jiu and H. Yu,
Decay of solutions to the three-dimensional generalized Navier-Stokes equations, Asymptot. Anal., 94 (2015), 105-124.
doi: 10.3233/ASY-151307. |
[12] |
T. Kato,
Strong $L^p$-solutions of the Navier-Stokes equation in $\mathbb{R}^m$, with applications to weak solutions, Math. Z., 187 (1984), 471-480.
doi: 10.1007/BF01174182. |
[13] |
I. Kukavica,
Space-time decay for solutions of the Navier-Stokes equations, Indiana Univ. Math. J., 50 (2001), 205-222.
doi: 10.1512/iumj.2001.50.2084. |
[14] |
I. Kukavica and J. J. Torres,
Weighted bounds for the velocity and the vorticity for the Navier-Stokes equations, Nonlinearity, 19 (2006), 293-303.
doi: 10.1088/0951-7715/19/2/003. |
[15] |
N. H. Katz and N. Pavlović,
A cheap Caffarelli-Kohn-Nirenberg inequality for the Navier-Stokes equation with hyperdissipation, Geom. Funct. Anal., 12 (2002), 355-379.
doi: 10.1007/s00039-002-8250-z. |
[16] |
H.-O. Kreiss, T. Hagstrom, J. Lorenz and P. Zingano,
Decay in time of incompressible flows, J. Math. Fluid Mech., 5 (2003), 231-244.
doi: 10.1007/s00021-003-0079-1. |
[17] |
J. Leray, Étude de diverses équations integrales non lineaires et de quelques problémes que pose l'hydrodynamique, Thèses de l'entre-deux-guerres, 142 (1933), 88pp. |
[18] |
J. Leray,
Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math., 63 (1934), 193-248.
doi: 10.1007/BF02547354. |
[19] |
P. Li and Z. Zhai,
Well-posedness and regularity of generalized Navier-Stokes equations in some critical $Q$-spaces, J. Funct. Anal., 259 (2010), 2457-2519.
doi: 10.1016/j.jfa.2010.07.013. |
[20] |
J.-L. Lions, Quelques Méthodes De Résolution Des Problémes Aux Limites Non Linéaires, Dunod, Gauthier-Villars, Paris, 1969. |
[21] |
Q. Liu and J. Zhao,
Global well-posedness for the generalized magneto-hydrodynamic equations in the critical Fourier-Herz spaces, J. Math. Anal. Appl., 420 (2014), 1301-1315.
doi: 10.1016/j.jmaa.2014.06.031. |
[22] |
Q. Liu, J. Zhao and S. Cui,
Existence and regularizing rate estimates of solutions to a generalized magneto-hydrodynamic system in pseudomeasure spaces, Ann. Mat. Pura Appl. (4), 191 (2012), 293-309.
doi: 10.1007/s10231-010-0184-8. |
[23] |
S. Liu, F. Wang and H. Zhao,
Global existence and asymptotics of solutions of the Cahn-Hilliard equation, J. Differential Equations, 238 (2007), 426-469.
doi: 10.1016/j.jde.2007.02.014. |
[24] |
C. Miao, B. Yuan and B. Zhang,
Well-posedness of the Cauchy problem for the fractional power dissipative equations, Nonlinear Anal., 68 (2008), 461-484.
doi: 10.1016/j.na.2006.11.011. |
[25] |
M. E. Schonbek,
$L^2$ decay for weak solutions of the Navier-Stokes equations, Arch. Ration. Mech. Anal., 88 (1985), 209-222.
doi: 10.1007/BF00752111. |
[26] |
M. E. Schonbek,
Large time behaviour of solutions to the Navier-Stokes equations, Comm. Partial Differential Equations, 11 (1986), 733-763.
doi: 10.1080/03605308608820443. |
[27] |
E. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, USA, 1970.
![]() ![]() |
[28] |
W. A. Strauss,
Decay and asymptotic for $u_tt-\Delta u=F(u)$, J. Funct. Anal., 2 (1968), 409-457.
doi: 10.1016/0022-1236(68)90004-9. |
[29] |
S. Weng,
Remarks on asymptotic behaviors of strong solutions to a viscous Boussinesq system, Math. Method Appl. Sci., 39 (2016), 4398-4418.
doi: 10.1002/mma.3868. |
[30] |
M. Wiegner,
Decay results for weak solutions of the Navier-Stokes equations on $\mathbb{R}^n$, J. London Math. Soc., 35 (1987), 303-313.
doi: 10.1112/jlms/s2-35.2.303. |
[31] |
J. Wu,
Generalized MHD equations, J. Differential Equations, 195 (2003), 284-312.
doi: 10.1016/j.jde.2003.07.007. |
[32] |
J. Wu,
Regularity criteria for the generalized MHD equations, Comm. Partial Differential Equations, 33 (2008), 285-306.
doi: 10.1080/03605300701382530. |
[33] |
Z. Ye,
Global well-posedness and decay results to 3D generalized viscous magnetohydrodynamic equations, Ann. Mat. Pura Appl. (4), 195 (2016), 1111-1121.
doi: 10.1007/s10231-015-0507-x. |
[34] |
Y. Zhou,
Regularity criteria for the generalized viscous MHD equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 491-505.
doi: 10.1016/j.anihpc.2006.03.014. |
[35] |
Y. Zhou,
A remark on the decay of solutions to the 3-D Navier-Stokes equations, Math. Methods Appl. Sci., 30 (2007), 1223-1229.
doi: 10.1002/mma.841. |
[36] |
Y. Zhou,
Asymptotic behaviour of the solutions to the 2D dissipative quasi-geostrophic flows, Nonlinearity, 21 (2008), 2061-2071.
doi: 10.1088/0951-7715/21/9/008. |
[1] |
Yang Liu. Global existence and exponential decay of strong solutions to the cauchy problem of 3D density-dependent Navier-Stokes equations with vacuum. Discrete and Continuous Dynamical Systems - B, 2021, 26 (3) : 1291-1303. doi: 10.3934/dcdsb.2020163 |
[2] |
Daoyuan Fang, Bin Han, Matthias Hieber. Local and global existence results for the Navier-Stokes equations in the rotational framework. Communications on Pure and Applied Analysis, 2015, 14 (2) : 609-622. doi: 10.3934/cpaa.2015.14.609 |
[3] |
Peixin Zhang, Jianwen Zhang, Junning Zhao. On the global existence of classical solutions for compressible Navier-Stokes equations with vacuum. Discrete and Continuous Dynamical Systems, 2016, 36 (2) : 1085-1103. doi: 10.3934/dcds.2016.36.1085 |
[4] |
Reinhard Racke, Jürgen Saal. Hyperbolic Navier-Stokes equations II: Global existence of small solutions. Evolution Equations and Control Theory, 2012, 1 (1) : 217-234. doi: 10.3934/eect.2012.1.217 |
[5] |
Takeshi Taniguchi. The existence and decay estimates of the solutions to $3$D stochastic Navier-Stokes equations with additive noise in an exterior domain. Discrete and Continuous Dynamical Systems, 2014, 34 (10) : 4323-4341. doi: 10.3934/dcds.2014.34.4323 |
[6] |
J. Huang, Marius Paicu. Decay estimates of global solution to 2D incompressible Navier-Stokes equations with variable viscosity. Discrete and Continuous Dynamical Systems, 2014, 34 (11) : 4647-4669. doi: 10.3934/dcds.2014.34.4647 |
[7] |
Xin Zhong. Global strong solution and exponential decay for nonhomogeneous Navier-Stokes and magnetohydrodynamic equations. Discrete and Continuous Dynamical Systems - B, 2021, 26 (7) : 3563-3578. doi: 10.3934/dcdsb.2020246 |
[8] |
Giovanni P. Galdi. Existence and uniqueness of time-periodic solutions to the Navier-Stokes equations in the whole plane. Discrete and Continuous Dynamical Systems - S, 2013, 6 (5) : 1237-1257. doi: 10.3934/dcdss.2013.6.1237 |
[9] |
G. Deugoué, T. Tachim Medjo. The Stochastic 3D globally modified Navier-Stokes equations: Existence, uniqueness and asymptotic behavior. Communications on Pure and Applied Analysis, 2018, 17 (6) : 2593-2621. doi: 10.3934/cpaa.2018123 |
[10] |
Yingshan Chen, Shijin Ding, Wenjun Wang. Global existence and time-decay estimates of solutions to the compressible Navier-Stokes-Smoluchowski equations. Discrete and Continuous Dynamical Systems, 2016, 36 (10) : 5287-5307. doi: 10.3934/dcds.2016032 |
[11] |
Yuming Qin, Lan Huang, Zhiyong Ma. Global existence and exponential stability in $H^4$ for the nonlinear compressible Navier-Stokes equations. Communications on Pure and Applied Analysis, 2009, 8 (6) : 1991-2012. doi: 10.3934/cpaa.2009.8.1991 |
[12] |
Joel Avrin. Global existence and regularity for the Lagrangian averaged Navier-Stokes equations with initial data in $H^{1//2}$. Communications on Pure and Applied Analysis, 2004, 3 (3) : 353-366. doi: 10.3934/cpaa.2004.3.353 |
[13] |
Stefano Scrobogna. Global existence and convergence of nondimensionalized incompressible Navier-Stokes equations in low Froude number regime. Discrete and Continuous Dynamical Systems, 2020, 40 (9) : 5471-5511. doi: 10.3934/dcds.2020235 |
[14] |
Michele Campiti, Giovanni P. Galdi, Matthias Hieber. Global existence of strong solutions for $2$-dimensional Navier-Stokes equations on exterior domains with growing data at infinity. Communications on Pure and Applied Analysis, 2014, 13 (4) : 1613-1627. doi: 10.3934/cpaa.2014.13.1613 |
[15] |
Jian-Guo Liu, Zhaoyun Zhang. Existence of global weak solutions of $ p $-Navier-Stokes equations. Discrete and Continuous Dynamical Systems - B, 2022, 27 (1) : 469-486. doi: 10.3934/dcdsb.2021051 |
[16] |
Jishan Fan, Yasuhide Fukumoto, Yong Zhou. Logarithmically improved regularity criteria for the generalized Navier-Stokes and related equations. Kinetic and Related Models, 2013, 6 (3) : 545-556. doi: 10.3934/krm.2013.6.545 |
[17] |
Zaihong Jiang, Li Li, Wenbo Lu. Existence of axisymmetric and homogeneous solutions of Navier-Stokes equations in cone regions. Discrete and Continuous Dynamical Systems - S, 2021, 14 (12) : 4231-4258. doi: 10.3934/dcdss.2021126 |
[18] |
Luigi C. Berselli. An elementary approach to the 3D Navier-Stokes equations with Navier boundary conditions: Existence and uniqueness of various classes of solutions in the flat boundary case.. Discrete and Continuous Dynamical Systems - S, 2010, 3 (2) : 199-219. doi: 10.3934/dcdss.2010.3.199 |
[19] |
Joanna Rencławowicz, Wojciech M. Zajączkowski. Global regular solutions to the Navier-Stokes equations with large flux. Conference Publications, 2011, 2011 (Special) : 1234-1243. doi: 10.3934/proc.2011.2011.1234 |
[20] |
Keyan Wang. On global regularity of incompressible Navier-Stokes equations in $\mathbf R^3$. Communications on Pure and Applied Analysis, 2009, 8 (3) : 1067-1072. doi: 10.3934/cpaa.2009.8.1067 |
2020 Impact Factor: 1.327
Tools
Metrics
Other articles
by authors
[Back to Top]